Blender 2.79 physics and simulation
Here is one approach, there are probably better ones

Turn off gravity

Add the object you want to attract objects to and give it a force field



Set the strength of the field to a negative value. and the shape to "Surface"

Make it a passive rigid body



enable rigid body collision and set bounciness to 0 and friction to 1



Add the objects you want to move and give them a rigid body type of active



If you animate this the objects will move to the body and stick.  You'll need to tune the force strength to get the speed you want.